CMB spectra and bispectra calculations: making the flat-sky approximation rigorous

TL;DR

This paper rigorously develops flat-sky approximations for cosmic microwave background spectra and bispectra, providing explicit accuracy estimates and simplifying cases, thereby improving computational efficiency in cosmological analyses.

Contribution

It introduces a controlled framework for flat-sky approximations of CMB spectra and bispectra, including explicit next-to-leading order calculations and alternative descriptions for better accuracy.

Findings

Flat-sky approximation is accurate at 1% level for spectra without ISW contribution.

Explicit next-to-leading order terms are computed for flat-sky approximations.

An alternative bispectrum description improves control over flat-sky approximation accuracy.

Abstract

This article constructs flat-sky approximations in a controlled way in the context of the cosmic microwave background observations for the computation of both spectra and bispectra. For angular spectra, it is explicitly shown that there exists a whole family of flat-sky approximations of similar accuracy for which the expression and amplitude of next to leading order terms can be explicitly computed. It is noted that in this context two limiting cases can be encountered for which the expressions can be further simplified. They correspond to cases where either the sources are localized in a narrow region (thin-shell approximation) or are slowly varying over a large distance (which leads to the so-called Limber approximation). Applying this to the calculation of the spectra it is shown that, as long as the late integrated Sachs-Wolfe contribution is neglected, the flat-sky approximation at leading order is accurate at 1% level for any multipole. Generalization of this construction scheme to the bispectra led to the introduction of an alternative description of the bispectra for which the flat-sky approximation is well controlled. This is not the case for the usual description of the bispectrum in terms of reduced bispectrum for which a flat-sky approximation is proposed but the next-to-leading order terms of which remain obscure.